Abstract: In arid or semi-arid regions, vegetation absorbs water through the nonlocal effects of roots. The paper establishes a mathematical model with nonlocal delay and Holling-II functional response function. By mathematical analysis, the conditions under which the vegetation-water model generates the Turing pattern are obtained. The spatial distributions of vegetation under different delays are obtained by numerical simulation. The simulation results show that delay in vegetation density presents a ``parabolic phenomenon", and delay can cause a change in the pattern structure. Specifically, as delay increases, a pattern is transformed from uniform distribution to nonuniform distribution. When the delay parameter is less than the threshold, vegetation density increases with the decrease of delay. On the contrary, the density of vegetation will increase with the increase of delay. Besides, the functional response term coefficient is positively correlated with vegetation density. The numerical simulation results show the influence of nonlocal delay and Holling-II functional response function on vegetation pattern, which provide a new theoretical basis for vegetation protection.

Key words: nonlocal delay, vegetation, desertification, pattern